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(1)

SCERT TELANGANA

(2)

For each and every conceptual understanding, a real life context with appropriate illustrations are given in the textbook. Try to understand the concept through keen reading of context along with observation of illustration.

While understanding the concepts through activities, some doubts may arise. Clarify those doubts by through discussion with your friends and teachers, understand the mathematical concepts without any doubts.

''Do this/Do these" exercises are given to test yourself, how far the concept has been understood. If you are facing any difficulty in solving problems in these exercises, you can clarify them by discussing with your teacher.

The problems given in "Try this/try these", can be solved by reasoning, thinking creatively and extensively. When you face difficulty in solving these problems, you can take the help of your friends and teachers.

The activities or discussion points given "Think & disicuss" have been given for extensive understanding of the concept by thinking critically. These activities should be solved by discussions with your fellow students and teachers.

Different typs of problems with different concepts discussed in the chapter are given in an "Exercise" given at the end of the concept/chapter. Try to solve these problems by yourself at home or leisure time in school.

The purpose of "Do this"/do these", and "Try this/try these" exercises is to solve problems in the presence of teacher only in the class itself.

Where ever the "project works" are given in the textbook, you should conduct them in groups. But the reports of project works should be submitted individually.

Try to solve the problems given as homework on the day itself. Clarify your doubts and make corrections also on the day itself by discussions with your teachers.

Try to collect more problems or make new problems on the concepts learnt and show them to your teachers and fellow students.

Try to collect more puzzles, games and interesting things related to mathematical concepts and share with your friends and teachers.

Do not confine mathematical conceptual understanding to only classroom. But, try to relate them with your suroundings outside the classroom.

Student must solve problems, give reasons and make proofs, be able to communicate mathematically, connect concepts to understand more concepts &

solve problems and able to represent in mathematics learning.

Whenever you face difficulty in achieving above competencies/skills/standards, you may take the help of your teachers.

SCERT

TELANGANA

(3)

dJ© 8 dm>

MATHEMATICS

CLASS - VIII

(MARATHI MEDIUM)

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àH$meH$

àH$meH$ àH$meH$

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Voc§JmZm gaH$ma, h¡Xam~mX Voc§JmZm gaH$ma, h¡Xam~mX Voc§JmZm gaH$ma, h¡Xam~mX Voc§JmZm gaH$ma, h¡Xam~mX Voc§JmZm gaH$ma, h¡Xam~mX

H$m`ÚmMm AmXa H$am e¡j{UH$ CÝZVr gmYm

hŠH$ {_idm ào_mZo dmJm

SCERT TELANGANA

(4)

(ii)

First Published 2013

New Impressions 2014, 2015, 2016, 2017, 2018,2019

All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means without the prior permission in writing of the publisher, nor be otherwise circulated in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.

The copy right holder of this book is the Director of School Education, Hyderabad, Telanga na.

We have used some photographs which are under creative common licence. They are acknowledged at later (page vii).

This Book has been printed on 70 G.S.M. Maplitho, Title Page 200 G.S.M. White Art Card

Voc§JmZm emgZmìXmao ‘mo’$V {dVaU 2019-20 Voc§JmZm emgZmìXmao ‘mo’$V {dVaU 2019-20Voc§JmZm emgZmìXmao ‘mo’$V {dVaU 2019-20 Voc§JmZm emgZmìXmao ‘mo’$V {dVaU 2019-20 Voc§JmZm emgZmìXmao ‘mo’$V {dVaU 2019-20

Printed in India

at the Telangana Govt. Text Book Press, Mint Compound, Hyderabad,

Telangana.

SCERT TELANGANA

(5)

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lr.H$mHw$bmda‘ amOo§Ð aoÈ>rlr.H$mHw$bmda‘ amOo§Ð aoÈ>r lr .VmVm 춧H$Q>am‘m Hw$‘malr .VmVm 춧H$Q>am‘m Hw$‘malr .VmVm 춧H$Q>am‘m Hw$‘malr .VmVm 춧H$Q>am‘m Hw$‘malr .VmVm 춧H$Q>am‘m Hw$‘ma Eg.E. ¶wnrEg,{Vå‘mnwa,Mm‘XmnoQ>, ZbJm|S>m {Oëhm

Eg.E. ¶wnrEg,{Vå‘mnwa,Mm‘XmnoQ>, ZbJm|S>m {OëhmEg.E. ¶wnrEg,{Vå‘mnwa,Mm‘XmnoQ>, ZbJm|S>m {Oëhm

Eg.E. ¶wnrEg,{Vå‘mnwa,Mm‘XmnoQ>, ZbJm|S>m {OëhmEg.E. ¶wnrEg,{Vå‘mnwa,Mm‘XmnoQ>, ZbJm|S>m {Oëhm ‘w.A. OoS>nrnrEMEg, ‘wbwJwS>r Zoëbmoa {Oëhm‘w.A. OoS>nrnrEMEg, ‘wbwJwS>r Zoëbmoa {Oëhm‘w.A. OoS>nrnrEMEg, ‘wbwJwS>r Zoëbmoa {Oëhm‘w.A. OoS>nrnrEMEg, ‘wbwJwS>r Zoëbmoa {Oëhm‘w.A. OoS>nrnrEMEg, ‘wbwJwS>r Zoëbmoa {Oëhm

lr B§Ða ‘mohZlr B§Ða ‘mohZlr B§Ða ‘mohZlr B§Ða ‘mohZlr B§Ða ‘mohZ lr ¶ed§V Hw$‘ma Xdolr ¶ed§V Hw$‘ma Xdolr ¶ed§V Hw$‘ma Xdolr ¶ed§V Hw$‘ma Xdolr ¶ed§V Hw$‘ma Xdo lr. hm{Zn nm{Zdmblr. hm{Zn nm{Zdmblr. hm{Zn nm{Zdmblr. hm{Zn nm{Zdmblr. hm{Zn nm{Zdmb lr. Am{ef M§Ðmlr. Am{ef M§Ðmlr. Am{ef M§Ðmlr. Am{ef M§Ðmlr. Am{ef M§Ðm {dÚm ^dZ gmogm`Q>r, [agmog© goÝQ>a, CX`nwa, amOñWmZ

lr.eaZ Jmonmblr.eaZ Jmonmblr.eaZ Jmonmblr.eaZ Jmonmblr.eaZ Jmonmb Hw$‘ E‘.AM©ZmHw$‘ E‘.AM©ZmHw$‘ E‘.AM©ZmHw$‘ E‘.AM©ZmHw$‘ E‘.AM©Zm lr nr.{Ma§Ordrlr nr.{Ma§Ordrlr nr.{Ma§Ordrlr nr.{Ma§Ordrlr nr.{Ma§Ordr gm§p»¶H$ {d^mJ {díd{dÚmb¶ h¡Ðm~mX

gm§p»¶H$ {d^mJ {díd{dÚmb¶ h¡Ðm~mXgm§p»¶H$ {d^mJ {díd{dÚmb¶ h¡Ðm~mX gm§p»¶H$ {d^mJ {díd{dÚmb¶ h¡Ðm~mX gm§p»¶H$ {d^mJ {díd{dÚmb¶ h¡Ðm~mX

lr. MwŠH$m am_æ`m lr. MwŠH$m am_æ`m lr. MwŠH$m am_æ`m lr. MwŠH$m am_æ`m lr. MwŠH$m am_æ`m

{ejUVk h¡Ðm~mX, Am§.à.

lr. S>m°. EM.Ho$. XodZ lr. S>m°. EM.Ho$. XodZ lr. S>m°. EM.Ho$. XodZ lr. S>m°. EM.Ho$. XodZ lr. S>m°. EM.Ho$. XodZ

{ejU gëbmJma, {dÚm ^dZ gmogm`Q>r, aogmog© goÝQ>a, CX`nwa, amOñWmZ

_w»` gëbmJma _w»` gëbmJma_w»` gëbmJma _w»` gëbmJma_w»` gëbmJma

boIH$

boIH$boIH$

boIH$boIH$

g§nmXH$$

g§nmXH$$

g§nmXH$$

g§nmXH$$

g§nmXH$$

e¡j{UH$ ghmæ`H$ JQ> gXñ`

e¡j{UH$ ghmæ`H$ JQ> gXñ`

e¡j{UH$ ghmæ`H$ JQ> gXñ`

e¡j{UH$ ghmæ`H$ JQ> gXñ`

e¡j{UH$ ghmæ`H$ JQ> gXñ`

g_Ýd`H$$

g_Ýd`H$$g_Ýd`H$$

g_Ýd`H$$g_Ýd`H$$

aoImQ>Z aoImQ>ZaoImQ>Z aoImQ>Z aoImQ>Z

lr.àem§V gmoZr lr.àem§V gmoZrlr.àem§V gmoZr

lr.àem§V gmoZrlr.àem§V gmoZr lr eoI e{H$b Ah‘X lr eoI e{H$b Ah‘X lr eoI e{H$b Ah‘X lr eoI e{H$b Ah‘X lr eoI e{H$b Ah‘X lr Eg.E‘. BH$am‘ lr Eg.E‘. BH$am‘ lr Eg.E‘. BH$am‘ lr Eg.E‘. BH$am‘ lr Eg.E‘. BH$am‘

{dÚm ^dZ gmogm`Q>r, [agmog© goÝQ>a, CX`nwa, amOñWmZ g_Ýd`H$ -

g_Ýd`H$ -g_Ýd`H$ -

g_Ýd`H$ -g_Ýd`H$ - lr.~r.Ama.OJXrída Jm¢S>, lr.~r.Ama.OJXrída Jm¢S>, lr.~r.Ama.OJXrída Jm¢S>, lr.~r.Ama.OJXrída Jm¢S>, lr.~r.Ama.OJXrída Jm¢S>, àmMm`©, emgH$s` AÜ`mnH$ {dÚmb`, Am{Xbm~mX AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ - lr.àem§V ^mo`a, lr.àem§V ^mo`a, lr.àem§V ^mo`a, lr.àem§V ^mo`a, lr.àem§V ^mo`a, EgE, Jìh©_|Q> JOrQ>oS> hm`ñHw$b Z§~a 1, Am{Xbm~mX AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ - lr.ZmJoe MZ_Zdma,lr.ZmJoe MZ_Zdma,lr.ZmJoe MZ_Zdma,lr.ZmJoe MZ_Zdma,lr.ZmJoe MZ_Zdma,EgE, PoS>nrEMEg, B§Ðdoëbr, {Oëhm Am{Xbm~mX AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ - lr.gw^mf _wgio, lr.gw^mf _wgio, lr.gw^mf _wgio, lr.gw^mf _wgio, lr.gw^mf _wgio, EgE, `wnrEg, _wVÝZwa, {Oëhm Am{Xbm~mX AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ -

AZwdmXH$ -AZwdmXH$ - lr.JUoeHw$_ma gw`©d§er, lr.JUoeHw$_ma gw`©d§er, lr.JUoeHw$_ma gw`©d§er, lr.JUoeHw$_ma gw`©d§er, lr.JUoeHw$_ma gw`©d§er, EgE, `wnrEg emgH$s` _amR>r emim, OwZo ~ñWmH$, Am{Xbm~mX

‘amR>r g§JUH MmcH -$ amOoe XmZH$m, Am{Xcm~mX

‘amR>r g§JUH MmcH -$ amOoe XmZH$m, Am{Xcm~mX‘amR>r g§JUH MmcH -$ amOoe XmZH$m, Am{Xcm~mX

‘amR>r g§JUH MmcH -$ amOoe XmZH$m, Am{Xcm~mX‘amR>r g§JUH MmcH -$ amOoe XmZH$m, Am{Xcm~mX

SCERT

TELANGANA

(6)

(iv)

{eúmU hr ‘mZdmg AkmZmVwZ ‘wº$ Am{U gm‘϶©dmZ ~Z{dUmar EH$ H$m¶©àUmcr Amho. {eúmUmMr àM§S> Aì¶dñWm AmoiIwZ gd© gmYmaUdmXr g‘OmZr XO}Xma {eúmUmMr VaVwX H$aʶmMm AJXr ñnï> CÔoemZo àmW{‘H$ {eúmUmMo gmd©{ÌH$ H$aʶmMr O~m~Xmar KoVcr. Xwgè¶m nm¶arV ‘mܶ{‘H$ {eúmUmcm gmd©{ÌH$ qH$dm gd©gmYmaU H$aʶmMr MmcZm {‘imcr. ‘mܶ{‘H$ AmdñWm R>iH$nUo {H«$¶m{ec J{UVmnmgwZ gwédmV Pmcr. Cƒ àmW{‘H$

AmdñWoÀ¶m Aä¶mgmn¶ªV {eñVnwU©H$ J{UVmMo Aܶ¶Z Amho. ¶m AdñWoV g‘ñ¶oMr Vm{H©$H$ {gÜXVm, n«‘o¶ B˶mXr n[aMrV Ho$co Amho. ˶mnmgwZ doJim EH$ Img {df¶ Omo BVa H$moU˶mhr {df¶mMm ghJm‘r Amho. Á¶mV {dMmagaUr Am{U n¥WH$aU Amho.

Amnë¶m Am§Y«àXoemVrc {dÚm϶m© ‘moR>çm CËgmhmZo Am[U AmZ§XmZo J{UV {eH$Vrc Aer ‘cm ImÌr dmQ>Vo.J[UVmg ˶mÀ¶m {OdUmVrc ‘hËdmMm A§J ~ZdwZ Am{U AW©nwU©

àíZ gmoS>dwZ ¶m nwñVH$m§À¶m dmMZmìXmao J{UVmMo ‘wc^wV AmH$ma g‘OwZ KoVrc. {eúmH$m§Zm Aä¶mgH«$‘mVrc AdKS> {df¶mH$S>o cúm doKwZ Am{U AܶmnU emómVrc Ñî¶ g‘OwZ KoUo Am{U R>iH$ {df¶mH$S>o cúm H|$ÐrH¥$V H$aʶmMr JaO Amho. Aä¶mgH«$‘mÀ¶m n[aUm‘

H$maH$, ì¶dhmamgmR>r g§{H$U© ImocrMo dmVmdaU, {eH${dUo Am{U {eH$Uo àUm{cV A˶§V

‘hËdmMo Amho. dJ© ImocrÀ¶m g§ñH¥$Vrcm g§dY©U H$éZ ˶mÀ¶m ‘ZmV YZmË‘H$ éMr R>gdwZ {OdZe¡crÀ¶m doJdoJù¶m g§^mdVoMm Am{U A{^àm¶mVrc ’$aH$ Am{U {OdZ ho kmZmMr VhmZ Amho. ho {eúmUmìXmao ˶m§À¶m ‘ZmV R>g{dco nm{hOo.

amÁ¶ Aä¶mgH«$‘ ’«o$‘ dH©$ (SCF 2011) Zo XmIdcocr J{UV {eH${dʶmÀ¶m gXa H$ënUog J{UVmÀ¶m XO}Xma H$mJXmda l‘H$m¡eë¶ {Z‘uV H$éZ amÁ¶mV J{UV {eH${dʶmgmR>r nmR>çnwñVH$ à‘mUmMr ‘m§S>Ur Ho$cr. gd© ^mdZm§Zm gmH$ma H$aʶmMm à¶ËZ nmR>çnwñVH$mZo Ho$cm Amho. amÁ¶ {dÚm n[aemoYZ g§ñWm (SCERT ) Amnë¶m amÁ¶m‘Yrc gd© {eúmH$d¥§X Á¶mZr ¶m nmR>çnwñVH$m§À¶m {dH$mgmgmR>r hmV^ma cmdcm Am{U nmR>çnwñVH$ {dH$mg g{‘VrÀ¶m ‘ohZVrMr àe§gm Ho$cr. ‘r {Oëhm{eúmUm{YH$mar, ‘§S>i {eúmUm{YH$mar Am{U

‘w»¶ {eúmH$mMm A{Ve¶ Am^mar AmhmoV. ‘r g§ñWm Am{U g§KQ>Zm ¶m§Mm gwÜXm Am^mar Amho, Á¶m§Zr ¶m nmR>çnwñVH$mÀ¶m {dH$mgmV doi {Xcm ‘r H$m¶m©c¶mMo {ZXoeH$ Am{U emco¶ {eúmUmMo g§MmcH$ Á¶m§Zr ¶m nmR>çnwñVH$mÀ¶m {dH$mgmgmR>r gmhæ¶Vm àYmZ Ho$cr.

˶m§Mo gwÜXm ‘r Am^mar Amho. ¶m H$m‘mcm {Za§Va n[al‘mZo XOm©dmT>{dë¶m ~Ôc Vw‘Mo Am^mar AmhmoV.Vw‘À¶m ñnï>rH$aUmMo Am{U gëë¶mMo Amåhr ‘Znwd©H$ ñdmJV H$aVmo.

Place : Hyderabad

Date : 03 December 2012

g§MmbH$, g§MmbH$, g§MmbH$, g§MmbH$, g§MmbH$,

Eg.gr.B©.Ama.Q>r., h¡X«m~mX Eg.gr.B©.Ama.Q>r., h¡X«m~mXEg.gr.B©.Ama.Q>r., h¡X«m~mX Eg.gr.B©.Ama.Q>r., h¡X«m~mX Eg.gr.B©.Ama.Q>r., h¡X«m~mX

SCERT

TELANGANA

(7)

amÁ¶ Aä¶mgH«$‘ \«o‘ dH©$ (SCF - 2011) À¶m AmYmamZo Am§Y« àXoe emgZmZo gd©

{df¶m§À¶m Aä¶mgH«$‘mMr COiUr H$aʶmMm {ZíM¶ Ho$bm. ˶m§À¶m ‘mJUrZwgma emioVrb {dÚm϶mªÀ¶m {OdZmV emioÀ¶m ~mhoarb {OdZmer Z³H$sM g§~§Y Agm¶bm nm{hOo.

{ejUmMm (RTE - 2009)h³H$m‘wio H$iwZ Ambo H$s emioV àdoe H$aUmè¶m à˶oH$

{dÚm϶mªg Amdí¶H$ àm{dʶ 14 dfm©À¶m d¶mÀ¶m nmVirda àmßV Pmbo nm{hOo. Z°eZb H$arHw$b‘ \«o‘ dH©$ 2005 À¶m AmYmamda n[a{MV Ho$bobm nmR>H«$‘ hm A{Ve¶ Amdí¶H$

Amho. {deof åhUOo J{UV hm {dkmZ ho ‘mܶ{‘H$ nmVir da amï´>r¶ ÑîQ>rH$moZmVwZ Amnë¶m {dÚm϶mªZm J{UVmMm Am{U {dkmZmMm ‘O~wV AmYma XoʶmMr Amdí¶H$Vm Amho.

XoemMr e³Vr hr ˶mÀ¶m bmoH$m§Zm O~m~Xma Am{U j‘VmnwU© ~Z{dʶm da AgVo Oo ˶m§À¶m JaOm ^J{dVo, ‘hËdmH$m§em Am{U à¶ËZ{eb Am¡ÚmoJrH$ gm‘mOmÀ¶m JaOoda AmYmarV AgVo. àmܶ{‘H$ d ‘mܶ{‘H$ {ejUmMm J{UVmÀ¶m {ejH$m§Zr 8 Vo 10 dJm©À¶m nmR>çH«$‘mMo Aܶ¶Z H$éZ ˶m§Mr nmíd©^w‘rÀ¶m {dñVma Am{U gImob nUm g‘OwZ KoUo Am{U ˶m {df¶mMo Cn¶moJOZ {dÚm϶mªìXmao àmW{‘H$ Am{U CÀMàmW{‘H$ AdñWoV H$éZ KoVbo nm{hOo.

J{UVmMm nmR>çH«$‘ VrZ AdñWoV åhUOoM àmW{‘H$ CÀM àmW{‘H$ hm aMZmËH$

Am{U ‘mJ‘mohr Zo àdoe Ho$bm Amho. nmR>çH«$‘ hm aMZmËH$ énmV àdoe emoYwZ H$mT>ʶmda Omoa {Xbobm Amho Am{U ‘wb^wV J{UVmÀ¶m Am{U ˶mMo ñnï>rH$aU {Xbobo Amho. {dÚm϶mªZm gh^mJr hmoʶmgmR>r àoaUm, MMm© Am{U dJm©V CËgmhmZo ^mJ KoʶmgmR>r ˶mg Odi H$aVo. AmOMo nmR>çH«$‘ Aä¶mgH«$‘mÀ¶m Am{U e¡j{UH$ nmÌVoÀ¶m AmYmamdéZ {b{hë¶m Jobo Amho. TS SCERT Zo V¶ma Ho$boë¶m Aä¶mgH«$‘mMr n³H$s COiUr H$éZ CX¶mg Ambo Amho.

nmR>çH«$‘ ghm joÌmV {d^mJë¶m Jobo Amho. Ogo g§»¶m nÜXV, A§H$J{UV, {~OJ{UV,

^w‘rVr, gm§p»¶H$s Am{U {ZX}em§H$ ^w‘rVr ¶m joÌmer g§~§YrV Agboë¶m {df¶mÀ¶gm {eH$dUr‘wio e¡j{UH$ nmÌVo‘Yrb àm{dʶ Ogo g‘ñ¶gm gmoS>{dUo, Vm{H©$H$ {dMma, J{UVmMo nañna g§~§Y ‘m{hVrMo {d{dY énmV B˶mXr {dH$mg hmoVmo. J{UVm‘wio Aܶ¶ZmMr {eñV Am{U X¡Z{XZ n[apñWVrV gwÜXm hmoVo. nmR>çnwñVH$m‘wio à¶ËZ OmñV àmYmݶVm Am{U ܶm¶mdéZ n[al‘mMr dmT> hmoVo. bhmZ JQ>mVrb Adga Amho. Am{U ho H$am. Am{U à¶ËZ H$am ¶m énmVrb AZw^mdmbm Amdí¶H$ Agbobo H¥$˶ dJm©Vrb n[apñWVrbm AmQ>mo³¶mV AmUʶmgmR>r {ejH$m§Mr ‘XV Oéar Amho.

¶m nwñVH$mMo H$mhr {deof d¡{eï>¶o Imbr {Xbobo Amho.

¶m nwñVH$mMo H$mhr {deof d¡{eï>¶o Imbr {Xbobo Amho.¶m nwñVH$mMo H$mhr {deof d¡{eï>¶o Imbr {Xbobo Amho.

¶m nwñVH$mMo H$mhr {deof d¡{eï>¶o Imbr {Xbobo Amho.

¶m nwñVH$mMo H$mhr {deof d¡{eï>¶o Imbr {Xbobo Amho.

* ¶m YS>çmMr aMZm doJdoJù¶m nÜXVrZo Ho$br Amho. Á¶m‘wio {dÚmWu gd© Aä¶mgH«$‘mÀ¶m joÌmV éMr KoD$Z. Aä¶mgmÀ¶m à˶oH$ nXmV éMr R>odVmo.

SCERT

TELANGANA

(8)

àdoe Ho$bobm Amho. {Za{Zamio à¶ËZ ñnîQ>rH$aUmVwZ {Z‘m©U hmoVmV. ‘m{hVr g‘PʶmgmR>r ì¶mIogmR>r, ì¶m»¶m Z Ho$bobr nXo Am{U ñdV…{gÜX Ambo Z{dZ g§~§Y H$mT>Uo ˶mÀ¶m à‘o¶ åhUVmV. hr pñdH$maboë¶m VËdmMo VmH$s©H$ n[aUm‘

AmhoV.

* à˶oH$ à‘o¶ nmhʶmgmR>r H$miOr KoVbr nm{hOo Or ¶m à‘o¶mÀ¶m {gÜXVobm gmo߶m[aVrZo g‘OʶmgmR>r d¥Ë¶mÀ¶m gmhmæ¶mZo àma§^rH$ VaVwX Ho$br Amho.

* {Za§Va AmH$bZ H$éZ qH$‘V R>a{dʶmMr H$m¶©àUmbr dñV§wÀ¶m à¶ËZ H$am, Am{U {dMma,MMm© Am{U {bhm ¶mImbr AmÀN>mXbobr Amho. Aem à˶oH$ dñVw§Mm eodQ>r Aä¶mg {Xbobm Amho. Á¶m‘wio {ejH$mg nwU© 15 YS>çmVrb ‘wbm§Mm Hw$ebVm ‘m{hV hmoVmo.

* g§nwU© nmR>çH«$‘ 15 YS>çmV {d^mJbobm Amho. Á¶m‘wio nmR>§¶memVwZ bhmZ bhmZ àíZmbm VH$m©bm EH$ÌH$éZ CËgmhmZo J{UV {eH$dVmV.

* g§{JV {MÌ, AmH¥$˶m, dmMVm ¶oUmam AjamMm AmH$ma {ZíMrVnUo ‘wbm§Zm nmR>çm§e pñdH$mamʶmg ‘XV {‘iVo.

YS>m (1) : n[a‘o¶ g§»¶m Á¶m g§»¶m nÜXVrÀ¶m joÌmV ¶oVmV. n[a‘o¶ g§»¶m {^ÝZmnmgwZ H$em doJù¶m AmhoV. ¶mÀ¶m g§~§Y H$iVom. n[a‘o¶ g§»¶oMo JwUY‘© CXmhUmÀ¶m ñnï>rH$aUm H$éZ MMm© H$aVm ¶oVo. ‘wbm§Zm g§»¶oaofoda n[a‘o¶ g§»¶m§Zm nmhʶmMm AdH$me {Xbm.n[a‘o¶

g§»¶m§Zm g§»¶gm aofoda Xem§emÀ¶m énmV àXe©Z Am{U CbQ> 6 ì¶m YS>çmV dJ© mA{U dJ©‘wi ‘wbm§Zm nwU© dJ©, dJ© g§»¶oMo JwUY‘© Am{U Ad¶d nÜXVrZo dJ©‘wi H$mT>Uo

^mJmH$mamÀ¶m nÜXVrZo H$mT>oU ho ‘wbm§Zm g‘O{dʶmÀ¶m AmnU à¶ËZ H$é. KZ Am{U KZénmg {d{dY CXmhaUmÀ¶m ñnï>rH$aUmdéZ MMm© H$am.

YS>m (2) (4) (11) Am{U (12) ‘ܶo {~OJ{UV Amho. ¶m YS>çmV aofr¶ g‘rH$aU EH$m MbmV Amho. empãXH$ énmVrb Agboë¶m g‘ñ¶m§Mm Mb ‘m{hV H$aʶmgmR>r Am{U à{VjonU nÜXVrZo {VÀ¶m qH$‘Vr H$mT>ʶmgmR>r EH$ AdH$me ‘wbm§Zm {Xbm. KmVm§H$ Am{U KmV ¶m YS>çmV H$mhr AbJmarX‘ bm ‘moR>çm g§»¶oÀ¶m KmVm§H$mÀ¶m énmV {b{hʶmgmR>r {Xbobo Amho. {d{dY CXmhaUmÀ¶m ñnï>rH$aUmZo KmVm§H$mÀ¶m {Z¶‘mMr MMm© H$aʶmV Ambr. ~¡OrH$ amerÀ¶m YS>çmV Am{U Ad¶d ~¡{OH$ g‘mZVm Oer (a + b)2 a2

+ 2ab + b2, (a + b) (a − b) a2 − b2 Am{U (x + a) (x + b) = x2 + (a + b)x + ab {d{dY qH$‘VrdéZ VnmgUrMr MMm© H$aʶmV Ambr ¶m énmMo ~¡{OH$ am{eMo Ad¶d nmS>Uo, EHw$U àíZ ‘wbm§Zm gamdmgmR>r {Xbo AmhoV.

(vi)

SCERT

TELANGANA

(9)

MH«$dmT> ì¶mOmÀ¶m gwÌmZo Cn¶moJOZ ¶m ~hþbwH$ VwbZm Am{U MMm© ¶m YS>çmV Amho.

YS>m (10) hm g‘ Am{U ì¶ñV à‘mU ˶m‘ܶo g‘à‘mU, ì¶ñV à‘mU Am{U {‘l à‘mU Mr {Z˶ {OdUmVrb doJdoJir g‘ñ¶m ¶mV Amho.

YS>m (15) ‘ܶo g§»¶mer Ioi {dÚm϶mªZm AbJmarX‘ Am{U g§»¶mÀ¶m H$mhr Z‘wݶmìXmao {Z¶‘ ‘m{hV H$aʶmMm AdH$me {Xbm Amho. ^mJmH$mamÀ¶m {Z¶‘mÀ¶m MM}déZ Z{dZ {Z¶‘ emoYwZ H$mT>bo Amho. A°nbg§»¶m Am{U H$moS>o ‘wbm§‘ܶo CËgmh {Z‘m©U hmoʶmgmR>r {Xbo Amho.

{dÚm϶mªZo ˶mÀ¶m ^modVr nm{hboë¶m gÑî¶ Am{U aMZm Ho$boë¶m AmH¥$VrÀ¶m àe§gm H$aʶmÀ¶m CÔoemZo ^w‘rVrMr MMm© Ho$br Amho. YS>m (3) ‘ܶo Mm¡H$moZmÀ¶m aMZogmR>r ˶mda H§o$ÐrH¥$V Ho$bo Amho. Z‘wݶmgmR>r gd© aMZoMo Z‘wݶmgmR>r ñnï>rH$aU Ho$bobr CXmhaUo KoVbr Amho.YS>m(13) 3D Mo 2D‘ܶo ÑmZ ¶m ‘ܶo {dÚm϶mªMo 3D ìXmao {d{dY AmH¥$˶m emoYʶmgmR>r nwaogm AdH$me {Xbm Amho.

‘m{hVr hmVmiʶmgmR>r joÌmV AmH¥$˶m Am{U AmboImÀ¶m nÜXVrZo {dÚm϶mªË¶mda ˶mÀ¶m g^modVmbr kmZmMr AmoiI H$éZ KoVo.

YS>m (7)‘ܶo V³Vm Am{U AmboImV V³Ë¶mÀ¶m gmhmæ¶mZo ‘m{hVr gm‘J«rMo dJuH$aU Am{U gm‘J«rbm dma§dmaVm AmboImV àX{e©V H$aUo gmR>r Ogo ñV§^mboI, ~hþ^wO g§{Md dma§dmaVm dH«$ H$mhr CXmhaU ‘ܶ, ‘ܶH$ Am{U ~hþbwH$mMr COiUr H$aʶmgmR>r {Xbr Amho. Am{U AdKS> àíZ gmoS>{dʶmMm n¶m©¶r nÜXVrMr MMm© Ho$br Amho.

eodQ>r YS>m (9),‘ܶo àV{b¶ AmH¥$VrÀ¶m n¥ð>^mJmMo joÌ\i g‘b§~ Mm¡H$moZ,dVw©i, dVw©hmH$ma [a¨J mA{U dVw©iI§S>, n¥ï>^mJmMo joÌ\i Am{U KVmMo KZ\i Am{U Am¶VmMo KZ\i B˶mXr ~Xb AmnU MMm© Ho$br Ho$di Mm§Jë¶m nmR>çH«$nwñVH$mMo MMm© Ho$br Mm§Jë¶m nmR>çH«$‘ nwñVmH$mMo CËnmXZ XO©oXma {ejUmMr ImÌr XoV. Mmb{dV ZmhrV.

{eH$Umè¶mMm gh^mJ H¥$˶ Am{U g‘ñ¶m ‘ܶo ˶mg ImÌrZo g‘OPwZ KoVbo nm{hOo.

åhUwZ Amem H$aVo H$s, {ejH$ {ejUmÀ¶m àU{b¶ énmdbr qH$dm Z‘wݶmV ~XbwZ ˶mnmgwZ Ho$di {Z˶ Aä¶mgmVrb àíZ gmoS>dwZ ˶m g§X^m©g g‘OwZ MmVw©¶mZo g‘ñ¶m§

gmoS>{dë¶m nm{hOoV.

SCERT

TELANGANA

(10)

(viii)

OmO© nmoë¶m OmO© nmoë¶m OmO© nmoë¶m OmO© nmoë¶m

OmO© nmoë¶m (1887 - 1985)

H$mhr OUm§Zr ¶m àíZm§~Ôc {dMma Ho$cm {H$, àíZ gmoS>{dʶmMr H$cm {eH$cocr Amho H$m H$mhr OUm§V Vo A§Jr Agcoco H$m¡eë¶ Amho. ‘aU nmdcoë¶m OmO© nmoë¶m§Zr ¶mMo à^mdemcr Am{U ñnï> énmV CÎma {Xco.

nmoë¶mMo OÝ‘ h§Jar BWo B.g. 1887 ‘ܶo Pmcm. Am{U ˶m§Zr J{UVmV H$mhr df© ˶m§Zr PwarM ‘Yrc ñdrg BÝgrQ>çwQ> Am°’$ Q>o³ZmcmoOr ‘ܶo {eH${dco.

˶m§Zr {c{hcoë¶m AZoH$ nwñVH$m§‘Yrc hmD$ Qw> gmëd Am¶ (1945) ¶mMm OmñV A{^‘mZ dmQ>cm. hr nwñVHo$ gw‘mao EH$ {‘cr¶Z à{V {dH$ë¶m Jocr. Am{U ˶mMo ^mfm§Va 17 ^mfoV Ho$co JocoV.

nmoë¶mMo gmoS>{dʶmMo Mma {gÜXm§V 1.

1.

1.

1.

1. g‘ñ¶m g‘PZo:g‘ñ¶m g‘PZo:g‘ñ¶m g‘PZo:g‘ñ¶m g‘PZo:g‘ñ¶m g‘PZo:

hm {gÜXYm§V d¡H$ënH$[a˶m dmQ>cm {H$ ˶mMm C„oI H$aʶmMr JaO ^mgcr Zmhr. {dÚm϶mªZm AZoH$Xma ¶m g‘ñ¶ocm gmoS>{dVm§Zm ˶mMm n[al‘mV AS>MUr {Z‘m©U Pmë¶m H$maU ˶m§Zm Vr ~amo~a nwU© nUo g‘Pcr Zmhr qH$dm ˶mMm ^mJ g‘Pcm Zmhr. {eúmH$m§Zr Ago àíZ {dÚm϶mªZm {dMmaco nm{hOoV Ogo.

Vwåhmcm àíZ gm§JVm§Zm ˶mVrc gd© eãX g‘Oco H$m¶? Oa Zmhr Va Vwåhr nwñVH$mÀ¶m ‘mJÀ¶m nmZmdarc gwMrV nhm. qH$dm eãXH$mofmV nhm {OWo Vwåhmcm AmT>iwZ ¶oVo. {VWo nhm. Vwåhr emoYʶmgmR>r H$m¶ {dMmaVm qH$dm XmI{dʶmgmR>r Vwåhr Vmo àíZ Vw‘À¶m ñdV:À¶m eãXmV nwÝhm gm§Jm. àíZ gm§JʶmMm Xþgam Cnm¶ Amho. H$mhr g§»¶oÀ¶m énmVrc CXmhaUm§Mm Vn{ec Vwåhr ‘m§Sw> eH$Vm Omo Vwåhmg àíZ gw~moY H$aʶmgmR>r ‘XV {‘iVo. Vwåhr {MÌ qH$dm AmH¥$VrMm {dMma Ho$cmV H$m Omo Vwåhmcm g‘ñ¶m g‘PʶmgmR>r Cn¶moJr nS>Vmo. n«íZmMo CÎma ¶oʶmgmR>r BWo n¶m©á ‘m{hVr Amho H$m¶? BWo H$mhr OmñVrMr ‘m{hVr Amho H$m? CÎmamgmR>r Vwåhmcm IamoIM H$m¶ JaO Amho.

2 . 2 .2 . 2 .

2 . ¶moOZm V¶ma H$aUo¶moOZm V¶ma H$aUo¶moOZm V¶ma H$aUo¶moOZm V¶ma H$aUo¶moOZm V¶ma H$aUo

EH$Xm ˶mg nwU©nUo AmoiIwZ ˶mg ^anwa n[al‘ cmdwZ g‘ñ¶m gmoS>{dʶmgmR>r EH$ ¶moOZm ~Z{dUo na§Vw Vwåhr Mmcw H$aʶmg Km~é ZH$m Vwåhr ~amo~a añ˶mda AmhmV.BWo g‘ñ¶m gmoS>{dʶmgmR>r nwîH$i ¶mo½¶

Cnm¶ AmhoV. ˶mMm à¶ËZ H$am Am{U ¶eñdr Cnm¶mZo hiwhiw H$mhr An¶eñdr MmMUrìXmao CX¶mg ¶oVo.

S>mdnoMmÀ¶m AnwU© ¶mXrV Amho.

* A§XmO Am{U VnmgUr * Z‘wʶmgmR>r nmhUo * H«$‘dmarZo ¶mXr ~Z{dUo * {MÌ H$mT>Uo *{deof [aVrZo gmoS>{dcoë¶m g‘ñ¶oMm {dMma AmYrM gmooS>{dcoë¶m gma»¶m g‘ñ¶oMm {dMma* g§ä¶mdVm H$mTw>Z Q>mH$Uo

* gmYr g‘ñ¶m gmoS>{dUo * g‘Vwë¶H$ àíZ gmoS>{dUo * gmå¶ Agcoë¶m g‘ñ¶m gmoS>{dUo. * g‘‘rVrMm dma * Z‘wʶmMm dmna * {deof g§X^© KoUo * ‘mJrc H$m‘ H$aUo* gai {dMma gaUr* gwÌmMm Cn¶moJ

* g‘rH$aU gmoS>{dUo* {dÜXdVm XmI{dUo 3 .

3 .3 . 3 .

3 . ¶moOZm A‘cmV AmUZo¶moOZm A‘cmV AmUZo¶moOZm A‘cmV AmUZo¶moOZm A‘cmV AmUZo¶moOZm A‘cmV AmUZo

¶moOZm V¶ma H$aʶmnoúmm ¶moOZog A‘cmV AmUZo Zoh‘r gmono OmVo. gm‘mݶVm gdm©g H$miOrnwd©H$

nUm Am{U ghZ{ecVm hr Amdí¶H$ Z¡nwʶo AmhoV. Oa ¶w³Vr Vm~S>Vmo~ H$m‘ Z Hoë¶mg ˶mg {MH$Qw>Z amhmdo. Var gwÜXm ¶wº$s H$m‘ Z Ho$ë¶mg ˶mg gmoSw>Z Z{dZ H$m¡eë¶mMo S>mdnoMmMm à¶ËZ H$éZ nhm.

MwH$sÀ¶m ‘mJm©cm OmD$ ZH$m ¶mM Cnm¶mZo J{UV, gmoS>{dë¶m OmVo.

4 . 4 .4 . 4 .

4 . ^wVH$mimH$S>o Ñï>rúmon H$aUo^wVH$mimH$S>o Ñï>rúmon H$aUo^wVH$mimH$S>o Ñï>rúmon H$aUo^wVH$mimH$S>o Ñï>rúmon H$aUo^wVH$mimH$S>o Ñï>rúmon H$aUo

Vw‘À¶m {dMmaUo Am{U {ZpíMVnUo n¥W¸$aU H$éZ CcJS>m nwU© H$aʶmgmR>r ^wVH$miH$mS>o Ñï>rnoúmm Ho$ë¶mg OmñV ¶e ’$º$ àíZ gmoS>{dʶmMr Jwé{H$„r H$m¶ Amho? ho AmnUmg J{UVmMr eº$s H$er àmá hmoVo. gm‘϶mªgmo~V Mm§Jco {dMma da ¶oD$Z g‘ñ¶m gmoS>{dʶmgmR>r ¶mMm H$Yr gm‘Zm hmoV Zmhr.

George Polya (1887-1985)

SCERT TELANGANA

(11)

YS>çmMo YS>çmMo YS>çmMo YS>çmMo

YS>çmMo gwMrgwMrgwMrgwMrgwMr Aä¶mgH«$‘ nwU© H$aʶmMmAä¶mgH«$‘ nwU© H$aʶmMmAä¶mgH«$‘ nwU© H$aʶmMmAä¶mgH«$‘ nwU© H$aʶmMmAä¶mgH«$‘ nwU© H$aʶmMm nmZ H«$‘m§H$nmZ H«$‘m§H$nmZ H«$‘m§H$nmZ H«$‘m§H$nmZ H«$‘m§H$

g§»¶m g§»¶m g§»¶m

g§»¶m g§»¶m H$mbmdYr H$mbmdYr H$mbmdYr H$mbmdYr H$mbmdYr

{df¶ gwMr

1 n[a‘o¶ g§»¶m OwZ 1 - 3 3

2 EH$m MbmVrb aofr¶ g‘rH$aUo OwZ/Owb¡ 3 4 - 5 8

3 Mm¡H$moUmMr aMZm Owb¡ 59-80

4 KmVm§H$ Am{U KmV Owb¡ 81-95

5 JwUmoËVamMm dmna H$éZ Am°JñQ> 96-123

ametMr VwbZm H$aUo

6 dJ©‘wi Am{U KZ‘wi Am°JñQ> 1 2 4 - 1 4 9

7 dma§dmaVm {dVaU gmaUr AmboI gßQ>|~a 1 5 0 - 1 8 2 8 ^w‘rVr¶ AmH¥$˶m§Mm emoY bmdUo gßQ>|~a/Am°³Q>m|~a 1 8 3 - 2 0 0 9 àVbr¶ AmH¥$˶m§Mo joÌ\i Am°³Q>m|~a 201-232

1 0 g‘ Am{U ì¶ñV à‘mU Zmoìh§§o~a 2 3 3 - 2 4 9

1 1 ~¡OrH$ nXmdbr {S>g§o~a 2 5 0 - 2 6 8

1 2 Ad¶d nmS>Uo {S>g§o~a 2 6 9 - 2 8 3

1 3 3 S>r Mo 2 S>r ‘ܶo ÑmZ OmZodmar 284-298 1 4 n¥ï>\i Am{U KZ\i OmZodmar/’o$~«wdmar 2 9 9 - 3 1 2

(KZ Am{U {XK©KZ)

1 5 g§»¶mer Ioi ’o$~«wdmar 313-338

16 COiUr ‘mM©

SCERT

TELANGANA

(12)

(x)

- aqdÐZmW Q>mJmoa - aqdÐZmW Q>mJmoa - aqdÐZmW Q>mJmoa - aqdÐZmW Q>mJmoa - aqdÐZmW Q>mJmoa

OZ JU ‘Z A{YZm`H$ O` ho

^maV ^m½` {dYmVm &

n§Om~, qgY, JwOamV, ‘amR>m Ðm{dS> CËH$b ~§J &&

qdÜ` {h‘mMb `‘wZm, J§Jm CÀN>b Ob{YVa§J &

Vd ew^ Zm‘o OmJo &

Vd ew^ Am{ef ‘mJo &

Jmho Vd O` JmWm OZ JU ԤJbXm`H$ O` ho

^maV ^m½` {dYmVm &

O` ho, O` ho, O` ho O` O` O` O` ho &

à{Vkm à{Vkm à{Vkm à{Vkm à{Vkm

- n¡[S>_au ì`§H$Q>m gwã~mamdw

^maV ‘mPm Xoe Amho. gmao ^maVr` ‘mPo ~m§Yd AmhoV. ‘mÂ`m Xoemda ‘mPo ào‘ Amho. ‘mÂ`m XoemVë`m g‘¥X²Y Am{U {d{dYVoZo ZQ>boë`m na§nam§Mm ‘bm A{^‘mZ Amho. Ë`m na§nam§Mm nmB©H$ hmoÊ`mMr nmÌVm ‘mÂ`m A§Jr `mdr, åhUyZ ‘r gX¡d à`ËZ H$arZ.‘r ‘mÂ`m nmbH$m§Mm, JwéOZm§Mm Am{U dS>rbYmè`m ‘mUgm§Mm ‘mZ R>odrZ Am{U àË`oH$mer gm¡OÝ`mZo dmJoZ. àmUr ‘mÌmda X¶m XmI{dU.

‘mPm Xoe Am{U ‘mPo Xoe~m§Yd `m§À`mer {Zð>m amIÊ`mMr ‘r à{Vkm H$arV Amho. Ë`m§Mo H$ë`mU Am{U Ë`m§Mr g‘¥X²Yr øm§VM ‘mPo gm¡»` gm‘mdbo Amho.

SCERT TELANGANA

(13)

n[a_o` g§»`m n[a_o` g§»`m n[a_o` g§»`m n[a_o` g§»`m n[a_o` g§»`m

1

1.0 àñVmdZmàñVmdZmàñVmdZmàñVmdZmàñVmdZm

àË`oH$s 5 én`o qH$_V Agboë`m 3 noZr gb_mbm {dH$V

¿`m`À`m hmoË`m. {VMm {_Ì gVre bm nU VgoM XmoZ noZr {dH$V ¿`m`À`m hmoË`m åhUwZ Vo XþH$mZmV Jobo. XþH$mZXmamZo gm§JrVbo H$s, 5 noZr Agboë`m nm°Ho$Q> Mr qH$_V 22 én`o Amho. Va EH$m noZrMr qH$_V {H$Vr Amho ? Vo AmnU ghOnUo _m{hV H$é eH$Vmo H$s àË`oH$ noZrMr qH$_V 225 én`o Amho hr qH$_V Xe©{dÊ`mgmR>r `oWo H$moUVr Var Z¡g{J©H$ g§»`m Amho H$m.? ho Xe©{dÊ`mgmR>r `oWo H$moUVr Var nwU© g§»`m qH$dm nwUmªH$ g§»`m Amho H$m ?

AOwZ EH$ CXmhaU nmhþ `m AOwZ EH$ CXmhaU nmhþ `mAOwZ EH$ CXmhaU nmhþ `m AOwZ EH$ CXmhaU nmhþ `m AOwZ EH$ CXmhaU nmhþ `m

{e_bm _Ü`o EH$m {d{eï> {Xder Zm|X Ho$boë`m Vmn_mZmMr _m{hVr Imbrb à_mUo Amho Ë`mMo {Z[ajU H$é.

doi g.10.00 dm. Xþ.12.00 dmOVm Xþ.3.00 dm. am.7.00 dm. am.10.00 dm.

Vmn‘mZ 11 oC 14 oC 17 oC 10 oC 5 oC

àË`oH$ g§X^m©V àVr Vmgmbm {H$Vr Vmn_mZ ~Xbbo ?

g§X^© I gH$miMo Vmg… à{V Vmgmg ~Xbobo Vmn_mZ 14 C 11 Co 2 o = 32 oC/hr.

gH$mir 10:00 Vo Xþnmar 12:00

g§X^© II XþnmaMo Vmg … à{V Vmgmg ~Xbobo Vmn_mZ 17 C 14 Co 3 o = 1 oC/hr.

Xþnmar 12:00 Vo Xþnmar 3:00

g§X^© III gm§`H$miMo Vmg … à{V Vmgmg ~Xbobo Vmn_mZ10 C 17 Co 4 o = 47 oC/hr.

Xþnmar 3:00 Vo gm`§H$mir 7:00

g§X^© IV amÌrMo Vmg … à{V Vmgmg ~Xbobo Vmn_mZ 5 C 10 Co 3 o = 35 oC/hr.

gm`§H$mir 7:00 Vo amÌr 10:00

darb g§X^m©V Amnë`mbm 32 oC, 1 oC, 47 oC, 35 oC `m g§»`m {_imë`m.

SCERT TELANGANA

(14)

`m Vmn_mZmV 32 oC, 1 oC, 47 oC, 35 oC `m g§»`m dmnaë`m Joë`m AmhoV. `m g§»`m§Zm H$m`

åhQ§>ë`m OmVo ?

`m n[a_mUmZm Xe©{dÊ`mgmR>r doJdoJi`m nÜXVrÀ`m g§»`mMr Amdí`H$Vm AgVo ho `oWo AmnU _m{hV Ho$bmo.

Aem nÜXVrÀ`m g§»`m {df`r MMm© H$é `m. 34, 79 , 1710, 32, 20132014, ...

EImXr g§»`m qp, À`m ñdénmV Xe©{dV Agob Am{U `oWo p Am{Uq nwUmªH$ AgVrb Am{U

q ≠ o, Va Aem g§»`m§Zm n[a_o` g§»`m åhUVmV. n[a_o` g§»`m ‘Q’Zo Xe©{dë`m OmVo. `m§Zm

^mJmH$ma g§»`m gwÜXm åhQ§>ë`m OmVo.

Imcr {Xcoë¶mMo {Z[ajU H$am.

Imcr {Xcoë¶mMo {Z[ajU H$am.Imcr {Xcoë¶mMo {Z[ajU H$am.

Imcr {Xcoë¶mMo {Z[ajU H$am.Imcr {Xcoë¶mMo {Z[ajU H$am.

H$moUË`mhr Z¡g{J©H$ g§»`mbm AmnU XmIdw eH$Vmo. CXm. 5 bm 51 qH$dm 102 qH$dm 152 ...

`mà_mUo H$moUË`mhr nwU© g§»`mZm AmnU XmIdw eH$Vmo.CXm. 0 bm 01 qH$dm 01 qH$dm 02 H$moUË`mhr nwUmªH$ g§»`mbm AmnU XmIdw eH$Vmo. CXm. -3 bm -31 qH$dm -62 darb {Z[ajUm déZ AmnU Agm {ZîH$f© H$mTw> eH$Vmo {H$ gd© Z¡g{J©H$ g§»`m, gd© nwU© g§»`m Am{U gd©

nwUmªH$ g§»`m `m n[a_o` g§»`m AmhoV.

ho H$am ho H$am ho H$am ho H$am ho H$am

1, 1

2 , −2, 0.5, 41

2, 33 7

, 0, 4

7, 0.3, 22, −5, 2

19, 0.125. ¶m g§»`mZm `mo½`

Ë`m {d^mJmV {bhm [EH$m g§»`obm EH$m nojm OmñV {d^mJmV {bhþ eH$Vm.]

(i) Z¡g{J©H$ g§»`m _ , ___ ___ ___ ___

(ii) nwU© g§»`m _ , ___ ___ ___ ___

(iii) nwUmªH$ _ , ___ ___ ___ ___ ___ ___ ___

(iv) n[a_o` g§»`m ___ ___ ___ ___ ___ ___ ___ ___

{Xboë`m g§»`m _Ü`o Vwåhr H$moUË`m Var g§»`obm n[a_o` g§»`oÀ`m {d^mJmVwZ ~mhoa R>odbo H$m ? àË`oH$ Z¡g{J©H$ g§»`m, nwU© g§»`m Am{U nwUmªH$ `m n[a_o` g§»`m Amho H$m ?

SCERT TELANGANA

(15)

à`ËZ H$am.

à`ËZ H$am.à`ËZ H$am.

à`ËZ H$am.

à`ËZ H$am.

1. h_rX åhUbm H$s, 53 hr n[a_o` g§»`m Amho Am{U 5 hr \ ŠV Z¡g{J©H$

g§»`m Amho. gmúmr åhUmbr XmoÝhr n[a_o` g§»`m AmhoV. Vwåhr H$moUmMo åhUUo _mÝ` H$aVm?

2. Imbrb dmŠ` gË` hmoÊ`mgmR>r EH$ CXmhaU Úm.

(i) gd© Z¡g{J©H$ g§»`m `m nwU© g§»`m AmhoV. nU gd© nwU© g§»`m Z¡g{J©H$ g§»`m hmoÊ`mMr JaO Zmhr.

(ii) gd© nwU© g§»`m ho nwUmªH$ AmhoV nU gd© nwUmªH$ ho nwU© g§»`m ZmhrV.

(iii) gd© nwUmªH$ ho n[a_o` g§»`m AmhoV nU gd© n[a_o` g§»`m nwUmªH$

hmoÊ`mMr JaO ZmhrV.

n[a_o` g§»`m da _wb^wV {H«$`m§Mm Aä`mg AmnU _mJrb dJm©V AJmoXaM Ho$boobm Amho.

1.2 n[a_o` g§»`mMo JwUY_©n[a_o` g§»`mMo JwUY_©n[a_o` g§»`mMo JwUY_©n[a_o` g§»`mMo JwUY_©n[a_o` g§»`mMo JwUY_©

1.2.1 g§d¥ËVg§d¥ËVg§d¥ËVg§d¥ËVg§d¥ËV:

(i) nwU© g§»`m Am{U nwUmªH$nwU© g§»`m Am{U nwUmªH$nwU© g§»`m Am{U nwUmªH$nwU© g§»`m Am{U nwUmªH$nwU© g§»`m Am{U nwUmªH$

nwU© g§»`m Am{U nwUmªH$ _Ü`o {H«$`m g§d¥ËV AgVmV. `mMr COiUr H$é `m.

`mo½` Aem CXmhaUm ìXmao Am{U Amdí`H$ Agboë`m H$maUmìXmao hm Imbrb VŠVm nwU© H$am.

g§»`m g§»`m g§»`m g§»`m

g§»`m {H«$`m{H«$`m{H«$`m{H«$`m{H«$`m

~oarO

~oarO~oarO

~oarO

~oarO dOm~mH$sdOm~mH$sdOm~mH$sdOm~mH$sdOm~mH$s JwUmH$maJwUmH$maJwUmH$maJwUmH$maJwUmH$ma ^mJmH$ma^mJmH$ma^mJmH$ma^mJmH$ma^mJmH$ma

nwU© g§»`m g§d¥ËV g§d¥ËV Zmhr g§d¥ËV g§d¥ËV Zmhr

H$moUË`mhr H$maU5−7 = −2 − − − H$maU 5 ÷8 = XmoZ a Am{U b −2 ho nwU© g§»`m$ 58 ho nwU© g§»`m nwU© g§»`mgmR>r Zmhr. − − − Zmhr

a + b ho nwU© g§»`m AgVo

nwUmªH$ g§d¥ËV a-b ho g§d¥ËV Zmhr nwUmªH$ AgVo − − − H$maU H$moUË`mhr XmoZ a

Am{U b nwUmªH$m gmR>r − − − − − −

− − −

Oa XmoZ nwU© g§»`m§Mr ~oarO nwU© g§»`m `oV Agob Va AmnU Ago åhUw eH$Vmo H$s, nwU© g§»`mMm

g§M ~oaOo _Ü`o g§d¥ËV JUwY_© AgVmo.

SCERT TELANGANA

(16)

(ii) n[a_o` g§»`m - g§d¥ËV {Z`_n[a_o` g§»`m - g§d¥ËV {Z`_n[a_o` g§»`m - g§d¥ËV {Z`_n[a_o` g§»`m - g§d¥ËV {Z`_n[a_o` g§»`m - g§d¥ËV {Z`_

(a) ~oarO~oarO~oarO~oarO~oarO

g_Om 27, 58 `m XmoZ n[a_o` g§»`m AmhoV. 2 57 8+ = 16 3556+ = 5156 Ambobo CËVa 5651 ho nwÝhm n[a_o` g§»`m Amho.

8 19 2

+ ⎜ = ______ ho n[a_o` g§»`m Amho H$m ?

2 2

7 7

+ = ___Vwåhmbm n[a_o` g§»`m {_imbr ? Imbrb OmoS>çm _Ü`o AOwZ H$mhr VnmgUr H$am.

5 1 7 2

3 , 0 ,

7 2 2 7

+ + +

Amnë`mbm Ago bjmV Ambo H$s, XmoZ n[a_o` g§»`mMr

~oarO nwÝhm n[a_o` g§»`mM Amho. åhUwZ ~oaOo _Ü`o n[a_o` g§»`m g§d¥ËV Amho. H$moUË`mhr XmoZ a Am{U b n[a_o` g§»`mgmR>r (a + b) ho n[a_o` g§»`mM AgVo.

åhUOoM ∀a, b ∈ Q ; (a + b) ∈ Q.

(b) dOm~mH$sdOm~mH$sdOm~mH$sdOm~mH$sdOm~mH$s:

g_Om 59 Am{U 34`m XmoZ n[a_o` g§»`m AmhoV.

Va 59 34 = 5 4 3 9× − ×36 = 2036 27 = 367

nwÝhm Amnë`mbm 367 hr n[a_o` g§»`m {_imbr (H$maU –7, 36 ho nwUmªH$ AmhoV Am{U 36 ho ewÝ` Zmhr Amho. åhUwZ 367 ho n[a_o` g§»`m Amho.).

Imbrb nwU© g§»`m§Zm gwÜXm `m gmR>r VnmgUr H$am.

(i) 2 3 14 9

3 7 21

= = _______________ ho n[a_o` g§»`m Amho H$m ?

(ii) 489 ⎞ − 1811 = _______________ ho n[a_o` g§»`m Amho H$m ?

H$moUË`mhr XmoZ n[a_o` g§»`mgmR>r, Ë`m§Mm \ aH$ gwÜXm n[a_o` g§»`mM Amho Ago AmnU _m{hV Ho$bmo. åhUwZ dOm~mH$s _Ü`o n[a_o` g§»`m H$moUË`mhr XmoZ a Am{U b n[a_o` g§»`m gmR>r a–b ho n[a_o` g§»`mM AgVo åUOoM ∀ a, b ∈ Q, (a–b) ∈Q

À¶mer g§~§Yr Amho gdm© gmR>r g‘Om A = {1, 2, 3}

3 hm KQ>H$ A ‘ܶo Amho Am{U Vo Xe©{dVo 3A Am{U AmnU ˶mbm 3 hm A er g§~§YrV Amho Ago dmMVmo.

{Z[ajU H$amx = 1 1 + 0 = 1 x = 2 2 + 0 = 2 x = 3 3 + 0 = 3 åhUOo gd© x A Va Amnë¶mbm x + 0 = x. ¶mbm AmnU Ago Xe©{dVmo H$s, x + 0 = x x A.¶mbm AmnU ¶mbm gdm© gmR>r åhUwZ dmMVmo. x A; x + 0 = x.

SCERT

TELANGANA

(17)

(c) JwUmH$maJwUmH$maJwUmH$maJwUmH$maJwUmH$ma

Imbrb Mo {Z[ajU H$am.

1 3 3× 2 = 2

6 11 66 33

5 2 10 5

× = =

3 5

7 × 2 = _________ ; 2 19

1 × 13 = ___________

darb gd© g§X^m©V Amnë`mbm Ago bjmV `oD$ eH$Vo H$s, XmoZ n[a_o` g§»`mMm JwUmH$ma n[a_o` g§»`mM AgVo. AmUIr H$mhr n[a_o` g§»`mÀ`m OmoS>çm gmR>r à`ËZ H$am Am{U VnmgUr H$am H$s Ë`mÀ`m JwUmH$ma n[a_o` g§»`m Amho qH$dm Zmhr. Aem XmoZ n[a‘o¶ g§»¶m Vwåhr ‘m{hV H$é eH$Vm H$m Á¶mMm JwUmH$ma n[a‘o¶ g§»¶m Zmhr.

JwUmH$mam_Ü`o n[a_o` g§»`m g§d¥ËV AgVmV ho AmnU _m[hV Ho$bmo AmhmoV.

H$moUË`mhr XmoZ n[a_o` g§»`m a Am{U b gmR>r a×b ho n[a_o` g§»`m Amho. ∀ a, b ∈ Q, a × b ∈Q (d) ^mJmH$ma^mJmH$ma^mJmH$ma^mJmH$ma^mJmH$ma

Imbrb XmoZ n[a_o` g§»`m Mo {Z[ajU H$am.

2 3, 7

8 g_Om 23 ÷ 78 = 23 × 87 = 1621 ho n[a_` g§»`m AmhoV. H$moUVo n[a_o`

g§»`m Amho.?

AOwZ XmoZ CXmhaUmgmR>r `mMr VnmgUr H$am.

5 5 2 5 1 5

7 ÷ 2 = 7 ÷ 1 = 7 × 2 = 14

2 6

3 11

÷ = __________ = _________ = __________

17 3 17

3÷ 13 = 1 ÷ 13 = ________ = ________ = ________

darb gdm© déZ Amnë`mbm Ago bjmV Ambo {H$, Oa AmnU XmoZ n[a_o` g§»`mMm ^mJmH$ma Ho$bm Va Amnë`mbm n[ao_o` g§»`m {_iVo. AmVm AmnU Ago åhUw eH$Vmo H$m, ^mJmH$mam _Ü`o n[a_o` g§»`m g§d¥ËV AgVmV. ?

ImbrbMr VnmgUr H$é `m: 0, 5 hr n[a_o` g§»`m AmhoV Am{U 50 Mr g§»`m H$aVm `oV Zmhr. nwU© g§»`mMm g_wh Q ho ^mJmH$ma _Ü`o g§d¥ËV Zmhr.

åhUwZ AmnU Ago åhUw eH$Vmo {H$, Oa AmnU Q _YwZ 0 H$mT>wZ Q>mH$bm Va Vo ^mJmH$ma _Ü`o g§d¥ËV hmoVo.

5

0 Mr 춻¶m H$m H$aVm ¶oV Zmhr?5

÷0 ^mJmH$ma H$am 0) 5 ? Vwåhr

^mJmH$ma nwU© H$é eH$Vm H$m?^mJmH$ma H$m`

Amho? EImÚm g§»`mbm ‘0’ Zo JwUmH$ma Ho$bo AgVm CËVa‘0’ho Vwåhr nm{hboM Agmb.åhUwZ ^mJmH$ma ho ewÝ` Zmhr Amho.

SCERT

TELANGANA

(18)

à`ËZ H$am à`ËZ H$amà`ËZ H$am à`ËZ H$am à`ËZ H$am

Oa AmnU nwUmªH$mÀ`m g§Mm_YwZ ewÝ` dJibm Va Vo ^mJmH$ma _Ü`o g§d¥ËV hmoV H$m ?

Z¡g{J©H$ g§»`m gmR>r AerM VnmgUr H$am.$

ho H$am ho H$am ho H$am ho H$am ho H$am

VŠË`m _Yrb [aH$mË`m OmJm ^am

g§»`m ~oarO dOm~mH$s JwUmH$ma ^mJmH$ma Z¡g{J©H$ g§»`m hmo`

nwU© g§»`m Zmhr

nwUmªH$ hmo`

n[a_o` g§»`m hmo`

1.2.2. H«$_{ZanojVoMm JwUY_© H«$_{ZanojVoMm JwUY_© H«$_{ZanojVoMm JwUY_© H«$_{ZanojVoMm JwUY_© H«$_{ZanojVoMm JwUY_© :

nwU© g§»`m Am{U nwUmªH$ `m XmoÝhr gmR>r doJdoJù`m {H«$`m H$éZ H«$_ {ZanojVoMm JwUY_m©Mr COiUr H$é

`m.

Imbrb VŠVm nwU© H$am.

(i) nwU© g§»`mnwU© g§»`mnwU© g§»`mnwU© g§»`mnwU© g§»`m

{H«$`m CXmhaU eoam

~oarO 2, 3 ho nwU© g§»`m AmhoV. ~oaOo _Ü`o

2+3 = 5 Am{U 3 + 2 = 5 W ho H«$_{Zanoj Amho.

∴ 2 + 3 = 3 + 2

dOm~mH$s 3 – 2 ho 2 –3À`m ~amo~a Amho H$m? dOm~mH$s ho H«$_{Zaonj Zmhr Amho.

JwUmH$ma --- ---

^mJmH$ma 4 ÷ 2 = ? 2 ÷ 4 = ? --- 4 ÷ 2 = 2 ÷ 4 Amho H$m ?

XomZ g§»`mÀ`m H«$_m_Ü`o ~Xb H$éZ pìXg§»`m_mZ {H«$`m Ho$ë`mZ§Va hr Ë`mÀ`m CËVam_Ü`o ~Xb hmoV ZgVo. `mbmM H«$_{ZanojVmMm JwUY_© åhUVmV.

a + b = b + a a × b = b × a

`oWo pìXg§»`m_mZ {H«$`m åhUOo Mma _wb^wV {H«$`m Ogo +, −, ×, ÷ n¡H$s EH$

g§d¥ËVmMm JwUY_©

SCERT TELANGANA

(19)

(ii) nwUmªH$nwUmªH$nwUmªH$nwUmªH$nwUmªH$

{H«$`m {H«$`m{H«$`m {H«$`m

{H«$`m CXmhaUCXmhaUCXmhaUCXmhaUCXmhaU eoameoameoameoameoam

~oarO --- ~oaOo_Ü`o nwUmªH$ H«$_{Zanoj Amho.

dOm~mH$s 2, 3 nwUm©§H$ AmhoV ...

2 − (3) = ? (3) − 2 = ? 2 − (3) = (3) − 2 = Amho H$m?

JwUmH$ma ... ... ... ... ... ...

^mJmH$ma ... ... ... ^mJmH$mam _Ü`o nwUm©§H$ ho H«$_{Zanoj Zmhr Amho.

(iii) n[a_o` g§»`mn[a_o` g§»`mn[a_o` g§»`mn[a_o` g§»`mn[a_o` g§»`m

(a) ~oarO~oarO~oarO~oarO~oarO

XmoZ n[a_o` 52, 43 ¿`m Am{U Ë`m§Mr ~oarO H$am.

( )3 2 5 1 ( )3

5 10 3 7

2 4 4 4 4

× + × −

+ = = =

Am{U( )43 + 52 = 1× − + ×( )34 2 5 = − +3 104 = 74 åhUwZ 52 + 43 = 43 + 52

n[a_o` g§»`mÀ`m BVa H$mhr OmoS>çmgmR>r hm {Z`_ VnmgwZ nhm.

1 5

2 + 7 Am{U 57 + 12 gmR>r, 12 + 57 = 57 + 12 hmoVo H$m ? ( )4

2 4 2

3 5 5 3 ?

+ = + hmoVo H$m ?

Aem n[a_o` g§»`mÀ`m OmoS>çm Vwåhr _m{hV H$é eH$Vm H$m Á`m§Mr ~oarO ~XbVo Ooìhm AmnU Ë`m§Mm H«$_ ~XbVmo ? aAm{Ub.XmoZ n[a_o` g§»`mgmR>r AmnU a + b = b + aåhUw eH$Vmo.

åhUwZ ~oaOo _Ü`o n[a_o` g§»`mMm g§M H«$_{Zanoj Amho.

∀ a, b ∈ Q, a + b = b + a

SCERT

TELANGANA

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